Goldblatt-Thomason Theorem

The Goldblatt-Thomason theorem characterizes which classes of frames are definable by modal formulas, providing a modal analogue of the classical preservation theorems in model theory.

Statement

Theorem8.2Goldblatt-Thomason Theorem

An elementary class of Kripke frames $\mathcal{C}$ is modally definable (i.e., $\mathcal{C} = \mathrm{Fr}(\Lambda)$ for some modal logic $\Lambda$ ) if and only if $\mathcal{C}$ is closed under:

Generated subframes: If $\mathcal{F} \in \mathcal{C}$ and $\mathcal{G}$ is generated from a point in $\mathcal{F}$ , then $\mathcal{G} \in \mathcal{C}$ .
Bounded morphic images (p-morphisms): If $\mathcal{F} \in \mathcal{C}$ and $f: \mathcal{F} \twoheadrightarrow \mathcal{G}$ is a bounded morphism, then $\mathcal{G} \in \mathcal{C}$ .
Disjoint unions: If $\{\mathcal{F}_i\}_{i \in I} \subseteq \mathcal{C}$ , then $\biguplus_{i \in I} \mathcal{F}_i \in \mathcal{C}$ .
Ultrafilter extensions: If $\mathcal{F} \in \mathcal{C}$ , then its ultrafilter extension $\mathcal{F}^{ue} \in \mathcal{C}$ .

Conversely, if $\mathcal{C}$ fails any of these, it is not modally definable (within elementary classes).

Key Notions

Definition8.10Bounded Morphism

A map $f: W_1 \to W_2$ between frames $(W_1, R_1)$ and $(W_2, R_2)$ is a bounded morphism (or p-morphism) if: (1) $w R_1 v \implies f(w) R_2 f(v)$ (preservation), and (2) $f(w) R_2 u \implies \exists v (w R_1 v \text{ and } f(v) = u)$ (back condition). Bounded morphisms preserve and reflect modal truth.

ExampleNon-Modally-Definable Class

The class of frames with exactly 3 worlds is elementary but not modally definable: it is not closed under disjoint unions (the disjoint union of two 3-world frames has 6 worlds). Hence no modal formula can define "having exactly 3 worlds."

RemarkSignificance

The Goldblatt-Thomason theorem parallels classical results: just as the Los-Tarski theorem characterizes first-order properties preserved under substructures, the Goldblatt-Thomason theorem characterizes first-order frame properties expressible in modal logic. It delineates the boundary of modal expressivity.

ExampleReflexivity is Modally Definable

The class of reflexive frames is elementary (defined by $\forall x \, xRx$ ) and modally definable (by the axiom $\Box p \to p$ ). One can verify closure under all four operations: generated subframes of reflexive frames are reflexive, bounded morphisms preserve reflexivity, disjoint unions of reflexive frames are reflexive, and ultrafilter extensions of reflexive frames are reflexive.